In physics and probability theory, mean field theory (MFT also known as self-consistent field theory) studies the behavior of large and complex stochastic models by studying a simpler model. Such models consider a large number of small individual components which interact with each other. The effect of all the other individuals on any given individual is approximated by a single averaged effect, thus reducing a many-body problem to a one-body problem.
Contents
- Formal approach
- Applications
- Ising Model
- Application to other systems
- Extension to time dependent mean fields
- References
The ideas first appeared in physics in the work of Pierre Curie and Pierre Weiss to describe phase transitions. Approaches inspired by these ideas have seen applications in epidemic models, queueing theory, computer network performance and game theory, as in the Quantal response equilibrium.
A many-body system with interactions is generally very difficult to solve exactly, except for extremely simple cases (random field theory, 1D Ising model). The n-body system is replaced by a 1-body problem with a chosen good external field. The external field replaces the interaction of all the other particles to an arbitrary particle. The great difficulty (e.g. when computing the partition function of the system) is the treatment of combinatorics generated by the interaction terms in the Hamiltonian when summing over all states. The goal of mean field theory is to resolve these combinatorial problems. MFT is known under a great many names and guises. Similar techniques include Bragg–Williams approximation, models on Bethe lattice, Landau theory, Pierre–Weiss approximation, Flory–Huggins solution theory, and Scheutjens–Fleer theory.
The main idea of MFT is to replace all interactions to any one body with an average or effective interaction, sometimes called a molecular field. This reduces any multi-body problem into an effective one-body problem. The ease of solving MFT problems means that some insight into the behavior of the system can be obtained at a relatively low cost.
In field theory, the Hamiltonian may be expanded in terms of the magnitude of fluctuations around the mean of the field. In this context, MFT can be viewed as the "zeroth-order" expansion of the Hamiltonian in fluctuations. Physically, this means an MFT system has no fluctuations, but this coincides with the idea that one is replacing all interactions with a "mean field". Quite often, in the formalism of fluctuations, MFT provides a convenient launch-point to studying first or second order fluctuations.
In general, dimensionality plays a strong role in determining whether a mean-field approach will work for any particular problem. In MFT, many interactions are replaced by one effective interaction. Then it naturally follows that if the field or particle exhibits many interactions in the original system, MFT will be more accurate for such a system. This is true in cases of high dimensionality, when the Hamiltonian includes long-range forces, or when the particles are extended (e.g., polymers). The Ginzburg criterion is the formal expression of how fluctuations render MFT a poor approximation, often depending upon the number of spatial dimensions in the system of interest.
While MFT arose primarily in the field of statistical mechanics, it has more recently been applied elsewhere, for example in inference, graphical models theory, neuroscience, and artificial intelligence.
Formal approach
The formal basis for mean field theory is the Bogoliubov inequality. This inequality states that the free energy of a system with Hamiltonian
has the following upper bound:
where
where
For the most common case that the target Hamiltonian contains only pairwise interactions, i.e.,
where
where
where
In order to minimize we take the derivative with respect to the single degree-of-freedom probabilities
where the mean field is given by
Applications
Mean field theory can be applied to a number of physical systems so as to study phenomena such as phase transitions.
Ising Model
Consider the Ising model on a
where the
Let us transform our spin variable by introducing the fluctuation from its mean value
where we define
The mean-field approximation consists of neglecting this second order fluctuation term. These fluctuations are enhanced at low dimensions, making MFT a better approximation for high dimensions.
Again, the summand can be reexpanded. In addition, we expect that the mean value of each spin is site-independent, since the Ising chain is translationally invariant. This yields
The summation over neighboring spins can be rewritten as
where
Substituting this Hamiltonian into the partition function, and solving the effective 1D problem, we obtain
where
We thus have two equations between
Application to other systems
Similarly, MFT can be applied to other types of Hamiltonian as in the following cases:
Extension to time-dependent mean fields
In mean-field theory, the mean field appearing in the single-site problem is a scalar or vectorial time-independent quantity. However, this need not always be the case: in a variant of mean-field theory called dynamical mean field theory (DMFT), the mean-field becomes a time-dependent quantity. For instance, DMFT can be applied to the Hubbard model to study the metal-Mott insulator transition.